Characterizing chain-compact and chain-finite topological semilattices

Банах, Тарас; Bardyla, Serhii

doi:10.1007/s00233-018-9921-x

“…In [3] Gutik studied and characterized e:sTS-closed topological groups (calling them H-closed topological groups in the class of semitopological semigroups). The papers [18,19] are devoted to recognizing C-closed topological semilattices for various categories C of topologized semigroups. In the paper [20] the author studied C-closedness properties in Abelian topological groups and proved the following characterization (implying the famous Prodanov-Stoyanov Theorem on the precompactness of minimal Abelian topological groups, see [20]).…”

Section: Problemmentioning

confidence: 99%

Categorically Closed Topological Groups

Банах

¹

2017

Axioms

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Let C be a category whose objects are semigroups with topology and morphisms are closed semigroup relations, in particular, continuous homomorphisms. An object X of the category C is called C-closed if for each morphism Φ ⊂ X × Y in the category C the image Φ(X) = {y ∈ Y : ∃x ∈ X (x, y) ∈ Φ} is closed in Y. In the paper we survey existing and new results on topological groups, which are C-closed for various categories C of topologized semigroups.

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“…In [16] Stepp proved that for any homomorphism h : X → Y from a chain-finite semilattice to a Hausdorff topological semilattice Y , the image h(X ) is closed in Y . In [1], the first authors improved this result of Stepp proving the following theorem.…”

Section: Introductionmentioning

confidence: 98%

“…In this paper we introduce a new cardinal invariant¯ (X ) of a Hausdorff topologized semilattice X , called the Lawson number of X . This was motivated by studying the closedness properties of complete topologized semilattices, see [1][2][3][4][5][6]. It turns out that complete semitopological semilattices share many common properties with compact topological semilattices, in particular their continuous homomorphic images in Hausdorff topological semilattices are closed.…”

Section: Introductionmentioning

confidence: 99%

The Lawson number of a semitopological semilattice

Банах

¹

,

Bardyla

²

,

Гутік

³

2021

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For a Hausdorff topologized semilattice X its Lawson number$$\bar{\Lambda }(X)$$ Λ ¯ ( X ) is the smallest cardinal $$\kappa $$ κ such that for any distinct points $$x,y\in X$$ x , y ∈ X there exists a family $$\mathcal U$$ U of closed neighborhoods of x in X such that $$|\mathcal U|\le \kappa $$ | U | ≤ κ and $$\bigcap \mathcal U$$ ⋂ U is a subsemilattice of X that does not contain y. It follows that $$\bar{\Lambda }(X)\le \bar{\psi }(X)$$ Λ ¯ ( X ) ≤ ψ ¯ ( X ) , where $$\bar{\psi }(X)$$ ψ ¯ ( X ) is the smallest cardinal $$\kappa $$ κ such that for any point $$x\in X$$ x ∈ X there exists a family $$\mathcal U$$ U of closed neighborhoods of x in X such that $$|\mathcal U|\le \kappa $$ | U | ≤ κ and $$\bigcap \mathcal U=\{x\}$$ ⋂ U = { x } . We prove that a compact Hausdorff semitopological semilattice X is Lawson (i.e., has a base of the topology consisting of subsemilattices) if and only if $$\bar{\Lambda }(X)=1$$ Λ ¯ ( X ) = 1 . Each Hausdorff topological semilattice X has Lawson number $$\bar{\Lambda }(X)\le \omega $$ Λ ¯ ( X ) ≤ ω . On the other hand, for any infinite cardinal $$\lambda $$ λ we construct a Hausdorff zero-dimensional semitopological semilattice X such that $$|X|=\lambda $$ | X | = λ and $$\bar{\Lambda }(X)=\bar{\psi }(X)=\mathrm {cf}(\lambda )$$ Λ ¯ ( X ) = ψ ¯ ( X ) = cf ( λ ) . A topologized semilattice X is called (i) $$\omega $$ ω -Lawson if $$\bar{\Lambda }(X)\le \omega $$ Λ ¯ ( X ) ≤ ω ; (ii) complete if each non-empty chain $$C\subseteq X$$ C ⊆ X has $$\inf C\in {\overline{C}}$$ inf C ∈ C ¯ and $$\sup C\in {\overline{C}}$$ sup C ∈ C ¯ . We prove that for any complete subsemilattice X of an $$\omega $$ ω -Lawson semitopological semilattice Y, the partial order $$\le _X=\{(x,y)\in X\times X:xy=x\}$$ ≤ X = { ( x , y ) ∈ X × X : x y = x } of X is closed in $$Y\times Y$$ Y × Y and hence X is closed in Y. This implies that for any continuous homomorphism $$h:X\rightarrow Y$$ h : X → Y from a complete topologized semilattice X to an $$\omega $$ ω -Lawson semitopological semilattice Y the image h(X) is closed in Y.

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“…The work of Velicko was continued by many topologists. Recently; authors in [2,3,4,5,6,7] have obtained several interesting results related to these sets. The notions of -closed sets and -open sets are introduced by Hdeib [8].…”

Section: Introductionmentioning

confidence: 99%

A new class between theta open sets and theta omega open sets

Ghour

¹

,

Al-Zoubi

²

2021

Heliyon

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We define  -closure operator as a new topological operator which lies between the -closure and the -closure. Some relationships between this new operator and each of -closure, -closure, and usual closure are obtained. Via  -closure operator, we introduce  -open sets as a new topology. Some mapping theorems related to the new topology are given. 2 topological spaces are characterized in terms of  -closure operator. Also, we use  -open sets to define  -regularity as a new separation axiom which lies strictly between -regularity and regularity. For a given topological space ( , ), we show that  -regularity is equivalent to the condition =  . Finally,  -continuity,  --continuity, weak  -continuity, and faint  -continuity are introduced and studied. Definition 2.1.[1] Let be a subset of a ts ( , ).a. The -closure of is denoted by ( ) and defined by

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Characterizing chain-compact and chain-finite topological semilattices

Cited by 25 publications

References 11 publications

Categorically Closed Topological Groups

Categorically Closed Topological Groups

The Lawson number of a semitopological semilattice

A new class between theta open sets and theta omega open sets

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